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Home , Persistent current, Superconductivity

Chapter 10:

References:

1. C. Kittel ; Introduction to state 2. M. Tinkham: Introduction to superconductivity 3. Paul Hansma, Tunneling Table of Contents (1)

(2)

(3)

Experimental Survey of Superconductivity Phenomenon WHAT IS A SUPERCONDUCTOR?

1. Zero resistance

2. Complete expulsion of

MH7699A.18 SUPERCONDUCTIVITY

Type of material What happens in a wire? Result

Conductor Collisions cause dissipation () flow easily (like water through a garden hose) No current flow at all

Electrons are tightly bound no flow (like a hose plugged with cement)

Superconductor No collisions No dissipation Electrons bind into pairs and No heat cannot collide No resistance (a frictionless hose) HOW SMALL IS THE RESISTANCE?

Copper Cylinder 1) Induce current 2) Current decays in about 1/1000 second

Superconducting Cylinder 1) Induce current 2) Current does not decay (less than 0.1% in a year) so, resistance is smaller than 1000 years by ──────────── 1/1000 second i.e., at least 1 trillion times! Why Superconductivity is so fascinating ?

 Fundamental SC mechanism

 Novel collective phenomenon at low temp

 Applications

Bulk: - , power storage - Magnetic - High field , MRI

Electronics: - SQUID - Josephson junction electronics POSSIBLE IMPACT OF SUPERCONDUCTIVITY

● - Superconductivity generators & motors - Power transmission & distribution - systems - for fusion power - Magnets for magneto-hydrodynamic power

● Transportation - Magnets for levitated trains - Electro-magnetic powered ships - Magnets for automobiles

● Health care - Magnetic resonance imaging

MH7699A.11 Low-Tc Superconductivity Mechanism 

Electron coupling k -k PROGRESS IN SUPERCONDUCTIVITY

200

180

160

140

120

100

80 N2

60 Transition (K) Transition Maximum Maximum superconducting 40

20 4.2K 23K 1911 1973 0 1900 1920 1940 1960 1980 2000 2020

Year A legacy of Superconductivity

Ted H. Geballe A legacy of Superconductivity

Bob Hammond PROGRESS IN SUPERCONDUCTIVITY

200 203K

180

160 155K 140

120 120K

100 90K 80 Liquid N2

60 Transition temperature (K) Transition Maximum Maximum superconducting 40 40K

23K 20 4.2K 1973 1911 0 1900 1920 1940 1960 1980 2000 2020

Year  The results are the work of Mikhail Eremets, Alexander Drozdov and their colleagues at the Max Planck Institute for Chemistry in Mainz, Germany last year. They find that when they subject samples of sulfide to extremely high pressures — around 1.5 million atmospheres (150 gigapascals) — and cool them below 203 K, the samples display the classic hallmarks of superconductivity: zero electrical resistance and a phenomenon known as the .

 Other hydrogen compounds may be good candidates for high- temperature superconductivity too. For instance, compounds that pair hydrogen with platinum, potassium, selenium or tellurium, instead of sulfur.  Zhang in Dallas and Yugui Yao of the Beijing Institute of in China predict that substituting 7.5% of the sulfur in with phosphorus and upping the pressure to 2.5 million atmospheres (250 GPa) could raise the superconducting transition temperature all the way to 280 K6, above water's point.

Experimental Electrical Resistivity of Metals

The electrical resistivity of most metals is dominated at room temperature (300K) by collisions of the conduction electrons with lattice and at liquid temperature (4K) by collisions with atoms and mechanical imperfections in the lattice (Fig. 11). Lattice phonons To a good approximation the rates are 1 1 1 Imperfections often independent. = + (46) And can be summed together τ τL τi

, where τL and τi are the collision times for scattering by phonons and by imperfections, respectively. The net resistivity is given by

r = rL + ri , Since r ~ 1/ t (47)

Often rL is independent of the number of defects when their concentration is small, and often ri is independent of temperature. This empirical observation expresses Matthiessen’s Rule.

Potassium metal

K

200

R/R At T > 

2 r  T 10

Different ri (0) but the same rL

resistance,

Figure 12 Resistance of potassium below 20 K, as measured on two specimens by D. MacDonald and K. Mendelssohn. The Relative different intercepts at 0 K are attributed to different concentrations of and static imperfections in the two specimens. Temperature, K. The temperature-dependent part of the electrical resistivity is proportional to the rate at which an collides with thermal phonons. One simple limit is at over the Debye temperature θ; here the phonon concentration is proportional to the temperature T, so that r  T for T > θ.

Nph  T , hence r  1/t  Nph  T The electrical resistivity of many metals and alloys drops suddenly to zero when the specimen is cooled to a sufficiently low temperature, often a tempera- ture in the range. This phenomenon, called superconductivity, was observed first by Kamerlingh Onnes in Leiden in 1911, three years after he first liquified helium. First SC found in Hg by 1911 !

R (ohm)

Figure 1 Resistance in ohms of a specimen of versus absolute temperature. The plot by Kamerlingh Onnes Marked the discovery of superconductivity.

Decay of persistent current from 1 year up to 105 year

Temp (K) Meissner Effect

It is an experimental fact that a bulk superconductor in a weak will act as a perfect diamagnet, with zero magnetic induction in the interior. When a specimen is placed in a magnetic field and is then cooled through the transition temperature for superconductivity, the magnetic flux originally present is ejected from the specimen. This is called the Meissner effect.

B= 0 Fundamental Mechanism

The superconducting state is an ordered state of the conduction electrons of the metal. Electron-Phonon Coupling

Cooper Pair formed by two electrons k, and –k with opposite spins near the Fermi level, as coupled through phonons of the lattice

The and origin of the ordering was explained by Bardeen, Cooper, and Schrieffer.3 BCS Theory, 1957

J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev. 106, 162 (1957); 108, 1175 (1957). The Discovery of Superconductivity • Early 90’s -- elemental SP metals like Hg, Pb, Al, Sn, Ga, etc. • Middle 90’s -- transitional metals, alloys, and

compounds like Nb, NbN, Nb3Sn, etc. • Late 90’s -- oxides

A-15 HTSC B1 A-15 compound A3B, with Tc = 15-23 K In the so called β–W structure With three perpendicular linear chains of A atoms on the cubic face, and B atoms are at body centered cubic site

1973 Nb3Ge, 23K ! Nb3Si ?? 3. The penetration depth and the length emerge as natural consequence of the BCS theory. The London equation is obtained for magnetic fields that vary slowly in space. Thus, the central phenomenon in superconductivity, the Meissner effect, is obtained in a natural way. penetration depth (λ) ; coherence length (ξ)

4. The electron density of orbitals D(EF) of one at the Fermi level, and the electron –lattice interaction U. For UD(EF) << 1, the BCS theory predicts:

2∆ /kBTc = 3.52 Where  is the Debye temperature and U is an attractive interaction (electron-phonon interaction).

For dirty metal (a poor conductor) → ρ(300)↑, U↑, Tc↑ (a good SC)

5. Magnetic flux through a superconducting ring is quantized and the effective unit of charge is 2e rather than e. Evidence of pairing of electrons Low temperature Superconductors

-- Mediated by Electron phonon coupling

-- McMillian formula for Tc

 : electron phonon coupling constant * : Coulomb repulsion of electrons

  N(0) < I2 >/ 2 Are electrons or phonons more important? The Phonon of the low Tc A-15 compound Nb3Al

Soft Phonons Can we raise the Tc higher than 30K?

Are we reaching the limitation of the BCS Theory ? Breakthrough in late 1986 By Bednorz and Muller

Start the HTSC Era ! 30K 40K 90K

120K 80K 40K High Temperature Superconductor YBa2Cu3O7 (90K)

CuO2 plane 2-D

Cu-O chain 1-D

Invention of Oxide Molecular Beam Epitaxy For HTSC Single Films. Will all non magnetic metal become SC at low T? (I) Destruction of Superconductivity by Magnetic Impurities It is important to eliminate from the specimen even trace quantities of foreign paramagnetic elements (II) Destruction of Superconductivity by Magnetic fields

At the critical temperature the critical field is zero: Hc(Tc)=0 Meissner Effect It is an experimental fact that a bulk superconductor in a weak magnetic field will act as a perfect diamagnet, with zero magnetic induction in the interior. When a specimen is placed in a magnetic field, and is then cooled through the transition temperature for superconductivity, the magnetic flux originally present is ejected from the specimen. This is called the Meissner effect.

B= 0 Meissner Effect M 1 B = Ba + 4휋M = 0 ; = － Ba 4휋 Eq.(1) Perfect

The magnetic properties cannot be accounted for by the assumption that a superconductor is a normal conductor with zero electrical resistivity. The result B = 0 cannot be derived from the characterization of a super- conductor as a medium of zero resistivity. From Ohm’s law, E = ρj, we see that if the resistivity ρ goes to zero while j is held finite, then E must be zero. By a Maxwell equation dB/dt is proportional to curl E, so that zero resistivity implies dB/dt = 0. This argument is not entirely transparent, but the result predicts that the flux through the metal cannot change on cooling through the transition. The Meissner effect contradicts this result and suggests that perfect diamagnetism is an essential property of the superconducting state. Type I superconductor Type II superconductor

Perfect Diamagnetism Type II Superconductors

1. A good type I superconductor excludes a magnetic field until superconductivity is destroyed suddenly, and then the field penetrates completely. 2. (a) A good type II superconductor excludes the field completely up to a field Hc1.

(b) Above Hc1 the field is partially excluded, but the specimen remains electrically superconducting.

(c) At a much higher field, Hc2, the flux penetrates completely and superconductivity vanishes. (d) An outer surface layer of the specimen may remain superconducting up to a still higher field Hc3. 3. An important difference in a type I and a type II superconductor is in the mean free path of the conduction electrons in the normal state. are type I, with κ < 1, will be type II. is the situation when κ = λ / ξ > 1. 1. A superconductor is type I if the surface energy is always positive as the magnetic field is increased, For H < Hc 2. And type II if the surface energy becomes negative as the magnetic field

is increased. For Hc1 < H < Hc2 The free energy of a bulk superconductor is increased when the magnetic field is expelled. However, a parallel field can penetrate a very thin film nearly uniformly (Fig. 17), only a part of the flux is expelled, and the energy of the superconducting film will increase only slowly as the external magnetic field is increased.

S N S N S Ginsburg Landau Parameter

k << 1 k >> 1 Type I Type II Vortex State In such mixed state, called the vortex state, the external magnetic field will penetrate the thin normal regions uniformly, and the field will also penetrate somewhat into the surrounding superconducting materials The term vortex state describes the circulation of superconducting currents in vortices throughout the bulk specimen,

Flux lattice Abrikosov triangular at 0.2K of NbSe lattice as imaged by 2 LT-STM, H. Hess et al

The vortex state is stable when the penetration of the applied field into the superconducting material causes the surface energy become negative. A type II superconductor is characterized bv a vortex state stable over a certain range of magnetic field strength; namely, between Hc1 and Hc2. Doping Pb with some In

Type I SC becomes type II SC Normal Core of Vortex 

k =  /  > 1 by pulsed magnetic field Chevrel (Ternary Sulfides)

Hc2

Hc2 vs T in A-15 compound

T Entropy S vs T for Aluminum

The small entropy change must mean that only a small fraction (of the order of 10-4) of the conduction electrons participate in the transition to the ordered superconducting state. Free energy vs T for Aluminum

dFN/ dT = dFS/ dT at TC

 FN =  FS at TC Zero , 2nd order

So that the phase transition is second order (there is no latent heat of transition at T c ). of an electron is 1 C = π2 D(ϵ ) k 2 T (34) el 3 F B

D(ϵF) = 3N/2ϵF = 3N/2 kBTF (35) 1 C = π2 Nk T/T . (36) el 2 B F Compare with CV = 2NkBT/TF where = k T TF is called the Fermi temperature, EF B F

K metal

1 γ = π2 Nk T/T Since ϵ  T  1/m ∴ γ  m (See Eq. 17) 2 B F F F At temperatures much below both the Debye temperature and the Fermi temperature, the heat capacity of metals may be written as the sum of electron and phonon contributions: C = γT + AT 3 C/T = γ + AT 2 (37) γ , called the Sommerfeld parameter At low T, the electronic term dominates. Heat Capacity of Ga at low T

Discontinuous change of

C at Tc, C/ Tc =1.43

Electronic part of heat capacity in SC state: Ces/γTc  a exp (-b Tc/T) proportional to -1/T, suggestive of excitation of electrons across an . Energy Gap

In a superconductor the important interaction is the electron-electron interaction via phonons, which orders the electron in the k space with respect to the of electrons.

The exponential factor in the electron heat capacity of a superconductor Is found to be –Eg/2kBT

Ces = γTc exp(-1.76 Tc/T)

The transition in zero magnetic field from the superconducting state to the normal state is observed to be a second-order phase transition.

Energy Gap of superconductors in Table 3

Eg(0)/kBTc = 3.52 Weak electron-phonon coupling

Eg(0)/kBTc > 3.52 Strong electron-phonon coupling

= 2 -4 Eg ~ 10 F

1-5 meV 3-10 eV

1/2 (T)/(0)= (1- T/Tc) Mean field theory

Sn Ta Nb

Eg(T) as the order parameter, goes smoothly to zero at Tc -- second order phase transition Effect

α~ 0.5

    M-1/2